How can I get HDRI shadows in an Eevee scene?
First, the reason for the lighting difference is that in the scene you provided, everything is lit by a World HDRI only. Eevee by default can only produce shadows from Light objects, so even if you turn the respectable options on, there won't be any indirect shadows visible. This is a technical limitation. Duarte in the linked answer elaborates on this, so there's no need to re-iterate everything here.

However, there is a quick solution to at least get a lot closer to the Cycles render: Adding an Irradiance Volume Light Probe allows you to bake the lighting solution into the scene dynamically.

To do that, in the 3D Viewport use Shift + A -> Light Probe -> Irradiance Volume.



Scale it up to encompass the areas of your scene which are visible through the camera. After that, in the Render Settings, use Indirect Lighting -> Bake Indirect Lighting. The scene will jitter a bit while the solution is baked. Afterwards, it should look something like this: